Systems Optimization Laboratory

Constrained Optimization

Professors Walter Murray
and Michael Saunders
lead the SOL research program on constrained optimization,
in close cooperation with Professor
Philip Gill
at UC San Diego.
Numerical optimization involves fundamental research on mathematical
methods for linear and nonlinear programming, as well as techniques
for implementing the methods as efficient and reliable computer
software. We have a special interest in algorithms for large-scale
problems. Our general-purpose packages (MINOS, NPSOL, SNOPT, etc.)
have been distributed to thousands of sites world-wide.
Feedback from users brings about many fruitful
collaborative efforts with industry, government and academia.

Algorithms

The algorithms implemented in SOL software include
the simplex method for linear programs (all solvers),
the reduced-gradient method (MINOS, for linear constraints),
a projected Lagrangian method (MINOS, for nonlinear constraints),
and SQP methods for linear and nonlinear constraints (NPSOL and SNOPT).

Active-set methods are employed for all problems or subproblems
involving linear constraints. These generate search directions
that stay on a certain subset of constraints (the working set).
The corresponding constraint gradients form the "working-set matrix".
This must be factorized to allow computation of search directions.
Most of the work per iteration lies in updating the matrix factors
and using them to compute the next search direction.

The dense solvers LSSOL, QPOPT and NPSOL employ orthogonal factors
of the working-set matrix for maximum stability.
For MINOS, SQOPT and SNOPT, the vital engine is
LUSOL:
a set of routines for computing sparse LU factors
of a square or rectangular matrix. New factorizations are
obtained using the Markowitz criterion to preserve sparsity,
and "Threshold Partial Pivoting", "Threshold Rook Pivoting",
or "Threshold Complete Pivoting" to preserve stability.
The LU factors are updated in a sparse and stable way by the
Bartels-Golub-Reid method.

Applications

SOL optimization software has been applied in many areas of
engineering, economics, finance, forestry, etc.
Example include: design of both yachts in the 1995
America's Cup final; online control of transmission networks for
electricity and gas; prediction of oil prices by the Federal Reserve;
climate modeling for the greenhouse debate; determination of forces on
the thigh bone prior to prosthesis insertion; trajectory optimization
for aircraft and spacecraft, including optimal control of the DC-X
experimental single-stage VTOL rocket.

Applications of NPSOL

America's Cup 1995 (Yacht Design)
NPSOL was used in different ways by both AC95 finalists.
For Team Dennis Conner (U.S.), NPSOL was used in conjunction with
Boeing's TRANAIR CFD system for optimal hull design of Young America.
For Team New Zealand, Andrew Philpott (University of Auckland) used NPSOL
to maximize the velocity around the course of 11 potential hull designs.
One design appeared significantly faster and was chosen to become
the New Zealand boat Black Magic.